Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

Abstract

We study the combinatorial properties associated with an earlier published, geometric algorithm capable of generating convex bodies in any primary equilibrium class (i.e. bodies with arbitrary numbers of equilibrium points) from a single ancestor. Primary equilibrium classes contain several topological secondary classes based on the arrangement of the equilibrium points. Here we show that the associated graph expansion algorithm is incomplete in the sense that using the same ancestor, not all secondary classes can be generated and we point out the nontrivial set of ancestors necessary to generate all secondary classes.